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19 <img class="icon32" alt="[\lambda\delta home]" title="\lambda\delta home" src="http://lambdadelta.info/images/crux_32.png" />
22 <div class="head1">cic:/matita/lambdadelta/basic_2/ (core λδ version 2)</div>
24 <img class="rule" alt="[Spacer]" title="\lambda\delta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
26 <div xmlns:ld="http://lambdadelta.info/" class="spacer">
29 <div xmlns:ld="http://lambdadelta.info/" class="text">
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33 <td class="snns capitalize italic sky">
34 <a href="http://lambdadelta.info/home.html">home</a>
36 <td class="snns capitalize italic magenta">
37 <a href="http://lambdadelta.info/news.html">news</a>
39 <td class="snns capitalize italic white">
40 <a href="http://lambdadelta.info/specification.html">specification</a>
42 <td class="snnn capitalize italic white">
45 <td class="snnn capitalize italic white">
48 <td class="snns capitalize italic orange">
49 <a href="http://lambdadelta.info/documentation.html">documentation</a>
51 <td class="snns capitalize italic green">
52 <a href="http://lambdadelta.info/implementation.html">implementation</a>
54 <td class="ssnn capitalize italic green">
59 <td class="snns capitalize sky">
60 <a href="http://lambdadelta.info/home.html#foreword">foreword</a>
62 <td class="snns capitalize magenta">
63 <a href="http://lambdadelta.info/news.html#milestones">milestones</a>
65 <td class="snns capitalize white">
66 <a href="http://lambdadelta.info/specification.html#v2">version 2</a>
68 <td class="snnn capitalize white">(<a href="http://lambdadelta.info/ground_2.html">background</a> - <a href="http://lambdadelta.info/basic_2.html">core</a> - <a href="http://lambdadelta.info/apps_2.html">applications</a>)</td>
69 <td class="snnn capitalize white">
72 <td class="snns capitalize orange">
73 <a href="http://lambdadelta.info/documentation.html#v2">version 2</a>
75 <td class="snns capitalize green">
76 <a href="http://lambdadelta.info/implementation.html#helena">helena</a>
78 <td class="ssnn capitalize green">
79 <a href="http://lambdadelta.info/osn/">Open Symbolic Notation (OSN)</a>
83 <td class="snss capitalize sky">
84 <a href="http://lambdadelta.info/home.html#citations">citations</a>
86 <td class="snss capitalize magenta">
87 <a href="http://lambdadelta.info/news.html#visibility">visibility</a>
89 <td class="snss capitalize white">
90 <a href="http://lambdadelta.info/specification.html#v1">version 1</a>
92 <td class="snsn capitalize white">(<a href="http://lambdadelta.info/ground_1.html">background</a> - <a href="http://lambdadelta.info/basic_1.html">core</a>)</td>
93 <td class="snsn capitalize white">(<a href="http://lambdadelta.info/static/matita/lambdadelta/">static HELM directory</a>)</td>
94 <td class="snss capitalize orange">
95 <a href="http://lambdadelta.info/documentation.html#v1">version 1</a>
97 <td class="snss capitalize green">
98 <a href="http://lambdadelta.info/implementation.html#lddl">library</a>
100 <td class="sssn capitalize green">(<a href="http://lambdadelta.info/static/lddl/">static LDDL directory</a>)</td>
106 <section>System's Syntax and Behavior</section>
107 <body>This is a summary of the "block structure"
108 of the System's syntactic items and reductions.
110 <table name="basic_2_blk"/>
111 <body>* In terms only.
112 ** In terms and local environments only.
113 *** In global environments only.
114 **** Sort level k in terms only.
117 <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="summary">Summary of the Specification <img class="icon37" alt="[butterfly]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b4.png" />
119 <div xmlns:ld="http://lambdadelta.info/" class="text">Here is a numerical account of the specification's contents
122 <div xmlns:ld="http://lambdadelta.info/" class="text">
123 <table cellpadding="4" cellspacing="0">
126 <td class="snns capitalize italic gray">category</td>
127 <td class="snns italic gray">objects</td>
128 <td class="snnn right italic gray">
131 <td class="snnn italic gray">
134 <td class="snnn right italic gray">
137 <td class="snnn italic gray">
140 <td class="ssnn right italic gray">
145 <td class="snns capitalize italic cyan">sizes</td>
146 <td class="snns italic cyan">files</td>
147 <td class="snnn right italic cyan">214</td>
148 <td class="snns italic cyan">characters</td>
149 <td class="snnn right italic cyan">202530</td>
150 <td class="snns italic cyan">nodes</td>
151 <td class="ssnn right italic cyan">1010247</td>
154 <td class="snns capitalize italic green">propositions</td>
155 <td class="snns italic green">theorems</td>
156 <td class="snnn right italic green">48</td>
157 <td class="snns italic green">lemmas</td>
158 <td class="snnn right italic green">675</td>
159 <td class="snns italic green">total</td>
160 <td class="ssnn right italic green">723</td>
163 <td class="snss capitalize italic yellow">concepts</td>
164 <td class="snss italic yellow">declared</td>
165 <td class="snsn right italic yellow">26</td>
166 <td class="snss italic yellow">defined</td>
167 <td class="snsn right italic yellow">46</td>
168 <td class="snss italic yellow">total</td>
169 <td class="sssn right italic yellow">72</td>
174 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="B">Stage "B"</div>
175 <ul xmlns:ld="http://lambdadelta.info/">
177 <span class="emph alpha">Ongoing.</span>
178 Context-sensitive subject equivalence
179 for native type assignment.
182 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A2">Stage "A2": "Extending the Applicability Condition"</div>
183 <ul xmlns:ld="http://lambdadelta.info/">
185 <span class="emph alpha">2017 January 17.</span>
186 Confluence for parallel r-transition on referred entries of local environments.
189 <ul xmlns:ld="http://lambdadelta.info/">
191 <span class="emph alpha">2016 September 15.</span>
192 Confluence for context-sensitive parallel r-transition on terms.
195 <ul xmlns:ld="http://lambdadelta.info/">
197 <span class="emph alpha">2016 April 16.</span>
198 Grammatical component reconstructed:
199 grammar, relocation, s_transition, s_computation, static
200 (anniversary milestone).
203 <ul xmlns:ld="http://lambdadelta.info/">
205 <span class="emph alpha">2016 March 25.</span>
206 Relocation with reference transforming maps (rtmap).
209 <ul xmlns:ld="http://lambdadelta.info/">
211 <span class="emph alpha">2015 October 9.</span>
212 λδ version 2A2 is started.
215 <div xmlns:ld="http://lambdadelta.info/" class="head3sn" id="A1">Stage "A1": "Extending the Applicability Condition"</div>
216 <ul xmlns:ld="http://lambdadelta.info/">
218 <span class="emph delta">2015 August 27.</span>
219 λδ version 2A1 appears too complex and is dismissed.
222 <ul xmlns:ld="http://lambdadelta.info/">
224 <span class="emph gamma">2014 October 28.</span>
225 λδ version 2A1 is released.
228 <ul xmlns:ld="http://lambdadelta.info/">
230 <span class="emph beta">2014 September 9.</span>
231 Iterated static type assignment defined (more elegantly)
232 as a primitive notion.
235 <ul xmlns:ld="http://lambdadelta.info/">
237 <span class="emph beta">2014 June 18.</span>
238 Preservation of stratified native validity
239 for context-sensitive computation on terms.
242 <ul xmlns:ld="http://lambdadelta.info/">
244 <span class="emph alpha">2014 June 9.</span>
245 Strong qrst-normalization
246 for simply typed terms.
249 <ul xmlns:ld="http://lambdadelta.info/">
251 <span class="emph alpha">2014 April 16.</span>
252 Lazy equivalence on local environments
253 added as q-step to rst-computation on closures
254 (anniversary milestone).
257 <ul xmlns:ld="http://lambdadelta.info/">
259 <span class="emph alpha">2014 January 20.</span>
260 Parametrized slicing of local environments
261 comprises both versions of this operation
262 (one from basic_1, the other used in basic_2 till now).
265 <ul xmlns:ld="http://lambdadelta.info/">
267 <span class="emph alpha">2013 August 7.</span>
268 Passive support for global environments.
271 <ul xmlns:ld="http://lambdadelta.info/">
273 <span class="emph alpha">2013 July 27.</span>
274 Reaxiomatized β-reductum as in rt-reduction.
277 <ul xmlns:ld="http://lambdadelta.info/">
279 <span class="emph alpha">2013 July 20.</span>
280 Context-sensitive strong rt-normalization
281 for simply typed terms.
284 <ul xmlns:ld="http://lambdadelta.info/">
286 <span class="emph alpha">2013 April 16.</span>
287 Reaxiomatized substitution and reduction
288 commute with respect to subclosure
289 (anniversary milestone).
292 <ul xmlns:ld="http://lambdadelta.info/">
294 <span class="emph alpha">2013 March 16.</span>
295 Mutual recursive preservation of stratified native validity
296 for rst-computation on closures.
299 <ul xmlns:ld="http://lambdadelta.info/">
301 <span class="emph alpha">2012 October 16.</span>
302 Confluence for context-free parallel reduction on closures.
305 <ul xmlns:ld="http://lambdadelta.info/">
307 <span class="emph alpha">2012 July 26.</span>
308 Term binders polarized to control ζ-reduction (not released).
311 <ul xmlns:ld="http://lambdadelta.info/">
313 <span class="emph alpha">2012 April 16.</span>
314 Context-sensitive subject equivalence
315 for atomic arity assignment
316 (anniversary milestone).
319 <ul xmlns:ld="http://lambdadelta.info/">
321 <span class="emph alpha">2012 March 15.</span>
322 Context-sensitive strong normalization
323 for simply typed terms.
326 <ul xmlns:ld="http://lambdadelta.info/">
328 <span class="emph alpha">2012 January 27.</span>
329 Support for abstract candidates of reducibility.
332 <ul xmlns:ld="http://lambdadelta.info/">
334 <span class="emph alpha">2011 September 21.</span>
335 Confluence for context-sensitive parallel reduction on terms.
338 <ul xmlns:ld="http://lambdadelta.info/">
340 <span class="emph alpha">2011 September 6.</span>
341 Confluence for context-free parallel reduction on terms.
344 <ul xmlns:ld="http://lambdadelta.info/">
346 <span class="emph alpha">2011 April 17.</span>
347 λδ version 2 is started.
350 <div xmlns:ld="http://lambdadelta.info/" class="head2dx" id="structure">Logical Structure of the Specification <img class="icon37" alt="[butterfly]" title="\lambda\delta butterfly" src="http://lambdadelta.info/images/b4.png" />
352 <div xmlns:ld="http://lambdadelta.info/" class="text">This table reports the specification's components and their planes.
354 <div xmlns:ld="http://lambdadelta.info/" class="text">
355 <table cellpadding="4" cellspacing="0">
358 <td class="snns top capitalize italic gray">component</td>
359 <td class="snns top italic gray">plane</td>
360 <td class="snns top gray">files</td>
361 <td class="ssnn top gray">
366 <td class="snns top capitalize italic cyan">rt-computation</td>
367 <td class="snns top italic cyan">uncounted context-sensitive rt-transition</td>
368 <td class="snns top cyan">cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )</td>
369 <td class="ssnn top cyan">
374 <td class="snns top capitalize italic water">rt-transition</td>
375 <td class="snns top italic water">parallel qrst-rtransition</td>
376 <td class="snns top water">fpb ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )</td>
377 <td class="ssnn top water">
382 <td class="nnns top capitalize italic water">
385 <td class="snns top italic water">t-bound context-sensitive rt-transition</td>
386 <td class="snns top water">lfpr ( ⦃?,?⦄ ⊢ ➡[?,?] ? )</td>
387 <td class="ssnn top water">lfpr_length lfpr_drops lfpr_fqup lfpr_frees lfpr_aaa lfpr_lfpx lfpr_lfpr</td>
390 <td class="nnns top capitalize italic water">
393 <td class="nnns top italic water">
396 <td class="snns top water">cpr ( ⦃?,?⦄ ⊢ ? ➡[?] ? )</td>
397 <td class="ssnn top water">cpr_drops</td>
400 <td class="nnns top capitalize italic water">
403 <td class="nnns top italic water">
406 <td class="snns top water">cpm ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )</td>
407 <td class="ssnn top water">cpm_simple cpm_drops cpm_lsubr cpm_lfxs cpm_cpx</td>
410 <td class="nnns top capitalize italic water">
413 <td class="snns top italic water">uncounted context-sensitive rt-transition</td>
414 <td class="snns top water">cnx ( ⦃?,?⦄ ⊢ ⬈[?,?] 𝐍⦃?⦄ )</td>
415 <td class="ssnn top water">cnx_simple cnx_drops</td>
418 <td class="nnns top capitalize italic water">
421 <td class="nnns top italic water">
424 <td class="snns top water">lfpx ( ⦃?,?⦄ ⊢ ⬈[?,?] ? )</td>
425 <td class="ssnn top water">lfpx_length lfpx_drops lfpx_fqup lfpx_frees lfpx_aaa</td>
428 <td class="nnns top capitalize italic water">
431 <td class="nnns top italic water">
434 <td class="snns top water">cpx ( ⦃?,?⦄ ⊢ ? ⬈[?] ? )</td>
435 <td class="ssnn top water">cpx_simple cpx_drops cpx_fqus cpx_lsubr cpx_lfxs cpx_lfdeq</td>
438 <td class="nnns top capitalize italic water">
441 <td class="snns top italic water">counted context-sensitive rt-transition</td>
442 <td class="snns top water">cpg ( ⦃?,?⦄ ⊢ ? ⬈[?,?] ? )</td>
443 <td class="ssnn top water">cpg_simple cpg_drops cpg_lsubr</td>
446 <td class="snns top capitalize italic green">static typing</td>
447 <td class="snns top italic green">atomic arity assignment</td>
448 <td class="snns top green">lsuba ( ? ⊢ ? ⫃⁝ ? )</td>
449 <td class="ssnn top green">lsuba_drops lsuba_lsubr lsuba_aaa lsuba_lsuba</td>
452 <td class="nnns top capitalize italic green">
455 <td class="nnns top italic green">
458 <td class="snns top green">aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )</td>
459 <td class="ssnn top green">aaa_drops aaa_fqus aaa_lfdeq aaa_aaa</td>
462 <td class="nnns top capitalize italic green">
465 <td class="snns top italic green">degree-based equivalence on referred entries</td>
466 <td class="snns top green">ffdeq ( ⦃?,?,?⦄ ≡[?,?] ⦃?,?,?⦄ )</td>
467 <td class="ssnn top green">ffdeq_fqup ffdeq_ffdeq</td>
470 <td class="nnns top capitalize italic green">
473 <td class="nnns top italic green">
476 <td class="snns top green">lfdeq ( ? ≡[?,?,?] ? )</td>
477 <td class="ssnn top green">lfdeq_length lfdeq_fqup lfdeq_lfdeq</td>
480 <td class="nnns top capitalize italic green">
483 <td class="snns top italic green">generic extension on referred entries</td>
484 <td class="snns top green">lfxs ( ? ⦻*[?,?] ? )</td>
485 <td class="ssnn top green">lfxs_length lfxs_drops lfxs_fqup lfxs_lfxs</td>
488 <td class="nnns top capitalize italic green">
491 <td class="snns top italic green">context-sensitive free variables</td>
492 <td class="snns top green">lsubf ( ⦃?,?⦄ ⫃𝐅* ⦃?,?⦄ )</td>
493 <td class="ssnn top green">lsubf_frees</td>
496 <td class="nnns top capitalize italic green">
499 <td class="nnns top italic green">
502 <td class="snns top green">frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )</td>
503 <td class="ssnn top green">frees_weight frees_drops frees_fqup frees_frees</td>
506 <td class="nnns top capitalize italic green">
509 <td class="snns top italic green">restricted ref. for local env.</td>
510 <td class="snns top green">lsubr ( ? ⫃ ? )</td>
511 <td class="ssnn top green">lsubr_length lsubr_drops lsubr_lsubr</td>
514 <td class="snns top capitalize italic grass">s-computation</td>
515 <td class="snns top italic grass">iterated structural successor for closures</td>
516 <td class="snns top grass">fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )</td>
517 <td class="ssnn top grass">fqus_weight fqus_drops fqus_fqup fqus_fqus</td>
520 <td class="nnns top capitalize italic grass">
523 <td class="nnns top italic grass">
526 <td class="snns top grass">fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )</td>
527 <td class="ssnn top grass">fqup_weight fqup_drops fqup_fqup</td>
530 <td class="snns top capitalize italic yellow">s-transition</td>
531 <td class="snns top italic yellow">structural successor for closures</td>
532 <td class="snns top yellow">fquq ( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )</td>
533 <td class="ssnn top yellow">fquq_length fquq_weight</td>
536 <td class="nnns top capitalize italic yellow">
539 <td class="nnns top italic yellow">
542 <td class="snns top yellow">fqu ( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )</td>
543 <td class="ssnn top yellow">fqu_length fqu_weight</td>
546 <td class="snns top capitalize italic orange">relocation</td>
547 <td class="snns top italic orange">generic slicing for local environments</td>
548 <td class="snns top orange">drops_vector ( ⬇*[?,?] ? ≡ ? ) ( ⬇*[?] ? ≡ ? )</td>
549 <td class="ssnn top orange">
554 <td class="nnns top capitalize italic orange">
557 <td class="nnns top italic orange">
560 <td class="snns top orange">drops ( ⬇*[?,?] ? ≡ ? ) ( ⬇*[?] ? ≡ ? )</td>
561 <td class="ssnn top orange">drops_lstar drops_weight drops_length drops_ceq drops_lexs drops_lreq drops_drops</td>
564 <td class="nnns top capitalize italic orange">
567 <td class="snns top italic orange">generic relocation for terms</td>
568 <td class="snns top orange">lifts_vector ( ⬆*[?] ? ≡ ? )</td>
569 <td class="ssnn top orange">lifts_lifts_vector</td>
572 <td class="nnns top capitalize italic orange">
575 <td class="nnns top italic orange">
578 <td class="snns top orange">lifts ( ⬆*[?] ? ≡ ? )</td>
579 <td class="ssnn top orange">lifts_simple lifts_weight lifts_tdeq lifts_lifts</td>
582 <td class="nnns top capitalize italic orange">
585 <td class="snns top italic orange">ranged equivalence for local environments</td>
586 <td class="snns top orange">lreq ( ? ≡[?] ? )</td>
587 <td class="ssnn top orange">lreq_length lreq_lreq</td>
590 <td class="nnns top capitalize italic orange">
593 <td class="snns top italic orange">generic entrywise extension</td>
594 <td class="snns top orange">lexs ( ? ⦻*[?,?,?] ? )</td>
595 <td class="ssnn top orange">lexs_length lexs_lexs</td>
598 <td class="snns top capitalize italic red">syntax</td>
599 <td class="snns top italic red">append for local environments</td>
600 <td class="snns top red">append ( ? @@ ? )</td>
601 <td class="ssnn top red">append_length</td>
604 <td class="nnns top capitalize italic red">
607 <td class="snns top italic red">degree-based equivalence for terms</td>
608 <td class="snns top red">deq ( ? ≡[?,?] ? ) </td>
609 <td class="ssnn top red">deq_deq</td>
612 <td class="nnns top capitalize italic red">
615 <td class="snns top italic red">same top term structure</td>
616 <td class="snns top red">tsts ( ? ≂ ? )</td>
617 <td class="ssnn top red">tsts_tsts tsts_vector</td>
620 <td class="nnns top capitalize italic red">
623 <td class="snns top italic red">closures</td>
624 <td class="snns top red">cl_weight ( ♯{?,?,?} )</td>
625 <td class="ssnn top red">
630 <td class="nnns top capitalize italic red">
633 <td class="nnns top italic red">
636 <td class="snns top red">cl_restricted_weight ( ♯{?,?} )</td>
637 <td class="ssnn top red">
642 <td class="nnns top capitalize italic red">
645 <td class="snns top italic red">global environments</td>
646 <td class="snns top red">genv</td>
647 <td class="ssnn top red">
652 <td class="nnns top capitalize italic red">
655 <td class="snns top italic red">local environments</td>
656 <td class="snns top red">lenv_length ( |?| )</td>
657 <td class="ssnn top red">
662 <td class="nnns top capitalize italic red">
665 <td class="nnns top italic red">
668 <td class="snns top red">lenv_weight ( ♯{?} )</td>
669 <td class="ssnn top red">
674 <td class="nnns top capitalize italic red">
677 <td class="nnns top italic red">
680 <td class="snns top red">lenv</td>
681 <td class="ssnn top red">
686 <td class="nnns top capitalize italic red">
689 <td class="snns top italic red">terms</td>
690 <td class="snns top red">term_vector ( Ⓐ?.? )</td>
691 <td class="ssnn top red">
696 <td class="nnns top capitalize italic red">
699 <td class="nnns top italic red">
702 <td class="snns top red">term_simple ( 𝐒⦃?⦄ )</td>
703 <td class="ssnn top red">
708 <td class="nnns top capitalize italic red">
711 <td class="nnns top italic red">
714 <td class="snns top red">term_weight ( ♯{?} )</td>
715 <td class="ssnn top red">
720 <td class="nnns top capitalize italic red">
723 <td class="nnns top italic red">
726 <td class="snns top red">term</td>
727 <td class="ssnn top red">
732 <td class="nnns top capitalize italic red">
735 <td class="snns top italic red">items</td>
736 <td class="snns top red">item_sd</td>
737 <td class="ssnn top red">
742 <td class="nnns top capitalize italic red">
745 <td class="nnns top italic red">
748 <td class="snns top red">item_sh</td>
749 <td class="ssnn top red">
754 <td class="nnns top capitalize italic red">
757 <td class="nnns top italic red">
760 <td class="snns top red">item</td>
761 <td class="ssnn top red">
766 <td class="nnss top capitalize italic red">
769 <td class="snss top italic red">atomic arities</td>
770 <td class="snss top red">aarity</td>
771 <td class="sssn top red">
779 <img class="rule" alt="[Spacer]" title="\lambda\delta rainbow rule" src="http://lambdadelta.info/images/rainbow.png" />
781 <div xmlns:ld="http://lambdadelta.info/" class="spacer">
784 <div xmlns:ld="http://lambdadelta.info/" class="spacer">
785 <a href="http://validator.w3.org/check?uri=referer">
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